Defining an Ellipse

Date: 07/16/2003 at 15:51:18
From: Chris
Subject: Defining an ellipse

A right cylinder of radius "r" (a lift duct) intercepts a plane (the 
deck of a hovercraft) at an angle of 30 degrees to the vertical.  How 
may the resulting ellipse be scribed? - i.e., how do I mark the hole 
to be cut in the deck?

Date: 07/16/2003 at 17:06:39
From: Doctor Peterson
Subject: Re: Defining an ellipse

Hi, Chris.

There are a couple of steps to the question: first, we have to find 
the dimensions of the ellipse, and then we have to draw it.

Here is a side view of your cylinder and plane; as I understand it, 
the deck is horizontal and the cylinder is tilted at a 30 degree 
angle to it:

                          -+
                      ----  \
               2r  ---       \
                ---           \
            ----               \
         --- A                  \
    ---+--------------------------------
       |\           2a            \
       | \                         \
       |  \                         \
       | A \                         \
       |    \                         \
       |     \                         \
       |      \                         \
               \                         \
                \                         \
                 \                        -+
                  \                   ----
                   \               ---
                    \           ---
                     \      ----
                      \  ---
                       +-

Here A is the angle (30 degrees), r is the radius of the cylinder (so 
that 2r is its diameter), and 2a is the long axis of the ellipse. 
Note that the short axis of the ellipse will be 2r. Using simple 
trigonometry,

  2r
  -- = cos(A)
  2a

so

  a = r/cos(A)

We call a the semi-major axis of the ellipse; the semi-minor axis 
is called b, so

  b = r

Now, how do we draw the ellipse? There are two best-known methods. 
One uses a pair of circles (with radii a and b) to plot points on the 
ellipse, and is described here:

   Accurate Drawing of an Ellipse
   http://mathforum.org/library/drmath/view/55085.html 

A quicker, but somewhat less accurate method, is given here:

   Ellipse Area and Circumference
   http://mathforum.org/library/drmath/view/55304.html 

and shown nicely here:

   Draw an ellipse, using pencil and string
   http://www.sciences.univ-nantes.fr/physique/perso/gtulloue/conics/
drawing/ellipse_string.html

The length of the string is our 2a, and the two nails or tacks (the 
foci) are a distance of 2c apart, where

  c^2 = a^2 - b^2

This fact may be seen here:

                         |
                    oooooBooooo-----------------
                oooo    /|\    oooo           ^
             ooo       / | \       ooo        |
           oo         /  |  \         oo      |
          o          /   |   \a         oo    b
        oo          /   b|    \           o   |
        o          /     |     \          o   |
       o          /      |      \          o  v
   ----o---------F-------O-------G---------A----
       o                 |   c             o
        o                |                o|
        oo               |                o|
          o              |              oo |
           oo            |            oo   |
             ooo         |         ooo     |
                oooo     |     oooo        |
                    ooooooooooo            |
                         |                 |
                         |<-------a------->|
                         |                 |

Since both the top point B and the side point A are drawn with the 
same total length of string, the lengths FB+GB = 2GB and FA+GA = 2OA 
are equal to 2a. This gives a right triangle GOB to which we can apply 
the Pythagorean theorem to get the fact I am demonstrating.

So you can measure the string and the nails carefully (which is 
tricky because of the stretchiness of the string), or use the two-
circle method to make a more precise, but more tedious, outline of 
the hole you need.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/